The result of eq:bernsteineq1 follows by applying Bernstein's inequality to \(\sum _{i=1}^n {\mathbf {x}}_i - \mathbb {E}[{\mathbf {x}}_i]\)  (so each summand is mean-zero). Per Bernstein's inequality, (cf. Theorem 2.8.1 in [1]}), there exists an absolute constant \(c > 0\)  such that for any \(s \ge 0\) ,
\(\mathbb {P}\left(\sum _{i=1}^n ({\mathbf {x}}_i - \mathbb {E}[{\mathbf {x}}_i]) \ge s\right) \le \exp \left(-c \min \left\lbrace \frac{s^2}{n\sigma ^2}, \frac{s}{\sigma } \right\rbrace  \right).\) 
